Question

In the exponential growth equation \(N_t=N_o e^{rt}\), e represents:

• A. The base of geometric logarithms
• B. The base of number logarithms
• C. The base of exponential logarithms
• D. The base of natural logarithms

Answer 1

The Correct Option is D

In the given exponential growth equation \(N_t=N_o e^{rt}\), the symbol \(e\) has a specific meaning. Let's break down the components of the equation and understand what each term represents:

• \(N_t\): The population size at time \(t\).
• \(N_o\): The initial population size.
• \(r\): The growth rate.
• \(t\): Time.
• \(e\): A mathematical constant approximately equal to 2.71828, which serves as the base of natural logarithms.

The purpose of \(e\) in this equation is to facilitate the modeling of continuous growth, which is commonly seen in natural processes. The natural logarithm base \(e\) is fundamental to the concept of exponential growth because it allows for the continuous, rather than discrete, compounding of growth rates.

Explanation of Options:

• The base of geometric logarithms: Geometric sequence logarithms are not typically based on \(e\), but on different bases that match the sequence.
• The base of number logarithms: This isn't a standard term in mathematics. Logarithms are usually described as natural (base \(e\)) or common (base 10).
• The base of exponential logarithms: This is ambiguous, as exponential growth and logarithms involve base \(e\), but this term is not commonly used or recognized in mathematics.
• The base of natural logarithms: Correct. Natural logarithms use \(e\) as their base, which is fundamental in continuous compound interest calculations and natural growth processes.

Based on the explanation above, the correct answer is: The base of natural logarithms.